Semantic Algebras. Semantic Algebras

Transcription

1 Semantic Algebras Wolfgang Schreiner Research Institute for Symbolic Computation (RISC-Linz) Johannes Kepler University, A-4040 Linz, Austria Wolfgang Schreiner RISC-Linz

2 Primitive Domains Truth values Domain Tr = B true: Tr false:tr not: Tr Tr or: Tr Tr Tr ( [] ): Tr D D D (for every domain D) Additional Nat operations equals: Nat Nat Tr lessthan: Nat Nat Tr greaterthan: Nat Nat Tr One element domain Domain Unit, the one element-domain (): Unit Wolfgang Schreiner 1

3 Primitive Domains Character strings Domain String = the character strings from elements of C (including error ) A, B, C,..., Z: String empty: String error: String concat: String String String length: String Nat substr: String Nat Nat String Computer store locations Domain Location, the address space in a computer store first-locn: Location next-locn: Location Location eq-locn: Location Location Tr less-l: Location Location Tr Wolfgang Schreiner 2

4 Product Domains Payroll information (name, payrate, hours) Domain Payroll = String Rat Rat newemp: String Payroll newemp(name) = (name, min, 0) where min Rat and 0 = makerat(0)(1) upd-payrate: Rat Payroll Payroll upd-payrate(pay, emp) = (emp 1, pay, emp 3) upd-hours: Rat Payroll Payroll upd-hours(hours, emp) = (emp 1,emp 2, addrat(hours)(emp 3)) compute-pay: Payroll Rat compute-pay(emp) = multrat(emp 2)(emp 3) (a 1, a 2,..., a n ) i = a i Wolfgang Schreiner 3

5 Sum Domains Revised payroll information Domain Payroll = String (Day + Night) Rat where Day = Night = Rat newemp: String Payroll newemp(n) = (n, inday(min), 0) move-to-day: Payroll Payroll move-to-day(emp) = (emp 1, cases emp 2 of isday(wage) inday(wage) isnight(wage) inday(wage), emp 3) compute-pay: Payroll Rat compute-pay(emp) = cases emp 2 of isday(wage) multrat(wage)(emp 3) isnight(wage) multrat(1.5) (multrat(wage)(emp 3)) Wolfgang Schreiner 4

6 Sum Domains Truth values as disjoint union Domain Tr = TT + FF where TT = Unit and FF = Unit true = intt() false = inff() not(t) = cases t of istt() inff() isff() intt() or(t, u) = cases t of istt() intt() isff() cases u of istt() intt() isff() inff() (t e [] f) = cases t of istt() e isff() f Wolfgang Schreiner 5

7 Sum Domains Finite lists Domain D = Unit + D + (D D) + (D (D D)) +... nil: D nil = inunit() cons: D D D cons(d,l) = cases l of isunit() ind(d) isd(y) ind D(d,y) isd D(y) ind (D D)(d,y)... hd: D D hd(l) = cases l of isunit() error isd(y) y isd D(y) fst(y) isd (D D)(y) fst(y)... Wolfgang Schreiner 6

8 Function Domains Dynamic arrays Domain Array = Nat A where A is a domain with an error element newarray: Array newarray = λn.error access: Nat Array A access(n, r) = r(n) update: Nat A Array Array access(n, v, r) = [n v]r Bounds are not restricted! access(m, update(n, v, r)) where m n = (update(n, v, r))(m) definition of access = ([n v]r)(m) definition of update = (λm.(equals m n) v [] r(m))(m) update = (equals m n) v [] r(m) function application = false v [] r(m) = r(m) Wolfgang Schreiner 7

9 Function Domains Dynamic arrays with curried operations Domain Array = Nat A where A is a domain with an error element newarray: Array newarray = λn.error access: Nat Array A access = λn.λr.r(n) update: Nat A Array Array access = λn.λv.λr.[n v]r Functions take one argument at a time! access(k) = λr.r(k) access(k)(r) r(k) (access k r) r(k) Wolfgang Schreiner 8

10 Lifted Domains Unsafe arrays of unsafe values Domain Uarr = Array where Array = Nat Tr and Tr = (B {error}) new-unsafe: Uarr new-unsafe = newarray access-unsafe: Nat Uarr Tr access-unsafe = λn.λr.(access n r) upd-unsf: Nat Tr Uarr Uarr upd-unsf = λn.λt.λr.(update n t r) Indices and elements may be improper! upd-unsf(plus one two)(not ( ))(new-unsafe) = upd-unsf(plus one two)(not ( ))(newarray) = upd-unsf(plus one two)(not ( ))(λn.error) = upd-unsf(three)( )(λn.error) = update(three)( )(λn.error) = [three ](λn.error) (not =λt.not(t)) Wolfgang Schreiner 9

11 Recursive Function Definitions Recursive function definitions need not define a function uniquely! q(x) = (equals x zero) one [] q(plus x one) f 1 (x) = f 2 (x) = one if x = zero otherwise one if x = zero two otherwise f 3 (x) = one How to formmalize that q yields the inted denotation f 1? (the problem will be handled later) (same with recursive domain definitions) Wolfgang Schreiner 10